The only time we really get a good clue is during EVA-01's first "berserk" incident. It would seem that Shinji was conscious inside of there, but pretty shaken up... the circuits were not connected, so he could not see anything, however he could probably feel that the EVA was moving (I mean, she flipped through the air and ran for a good distance!)
The only time that the pilot became absorbed was during EVA-01's fight with Zeruel. Well, in the actual series, anyway. A similar event happened to every person that became an EVA's soul during its "Contact Experiment".